It was the first time in years that protesters have been able to reach the US embassy, which is sheltered behind a series of checkpoints in the Iraqi capital's high-security Green Zone.

Hundreds of people in military fatigues marched through those checkpoints and broke into the embassy's visitor reception building, sparking fury from President Donald Trump and other top US officials.

Angered by US air strikes that killed two dozen fighters, they waved flags of Shiite paramilitary forces backed by Tehran and posters reading: "Parliament should oust US troops, or else we will!"

The bulk of protesters remained as night fell, setting up tents and beginning to cook dinner, signalling a possibly prolonged sit-in.

Demonstrators had scrawled "No to America!" and "Soleimani is my commander" on the embassy walls, referring to Tehran's pointman for Iraq, Major General Qasem Soleimani of the Revolutionary Guard Corps.

Trump blamed Iran and demanded Iraqi government forces protect the embassy compound.

"Iran is orchestrating an attack on the US Embassy in Iraq. They will be held fully responsible," he said via Twitter.

But Tehran hit back, slamming Washington for the "surprising audacity" in blaming Iran for the violence.

- 'First lesson' to US -

After his tweets, Trump called Iraq's caretaker premier Adel Abdel Madi to emphasise the need to protect US personnel and facilities in Iraq, the White House said.

Secretary of State Mike Pompeo had earlier called Abdel Mahdi and President Barham Saleh to pressure them to boost embassy security.

Pompeo "made clear the United States will protect and defend its people, who are there to support a sovereign and independent Iraq," the State Department said.

The Pentagon announced it was sending additional units to the embassy, and helicopters carrying US Marines from neighbouring Kuwait landed at the embassy late Tuesday, a security source told AFP.

But there were no plans to evacuate the mission and the ambassador, who had been on holiday leave, was on his way back to the embassy, US source said.

The assault came two days after US air strikes killed at least 25 fighters from the hardline Kataeb Hezbollah (Hezbollah Brigades), a component of the Hashed al-Shaabi military network.

The Pentagon said the strikes were in retaliation for a rocket attack by the group which killed an American contractor in Kirkuk.

It was the first US casualty from a spate of rocket attacks in recent weeks.

"Iran killed an American contractor, wounding many. We strongly responded, and always will," Trump tweeted Tuesday.

In response, Kataeb Hezbollah called for a funeral march to the US embassy.

Late Tuesday, the faction hailed the protest outside as a "first lesson" to Washington, "so that Trump knows he did something extremely stupid".

"The second lesson will be the law, which will oust the occupation forces," the force said in a statement.

- US warns Iran -

Dozens of Iraqi lawmakers have called on the government to review an agreement allowing some 5,200 US troops in Iraq, saying the strikes violated the country's sovereignty.

Those troops and other US targets have been hit in recent months by around a dozen rocket attacks, with American officials pointing the finger at Tehran-backed factions.

Many of those groups are linked to the powerful Hashed.

Hashed chief Faleh al-Fayyadh -- who is also Iraq's National Security Adviser -- was among the angry crowds attacking the embassy on Tuesday, as was the country's interior minister and other Hashed leaders.

Demonstrators appeared to gain easy access through the Green Zone to the embassy, throwing rocks, tearing down security cameras and setting a sentry box ablaze in anger over the US strikes.

The Hashed said at least 62 people were wounded after US embassy guards fired an initial volley of bullets, then switched to tear gas and stun grenades to disperse crowds.

The mob attack put a focus on the strains in the US-Iraqi relationship as allies of Iran, which enjoys significant support in parts of the Iraqi government, increasingly challenge Washington's influence in the country.

"This may well be the low point of US policy in Iraq," said Iraq expert Fanar Haddad of Singapore University's Middle East Institute.

Following the deadly strikes, the Iraqi government announced three days of mourning and threatened to summon the US ambassador.

"American forces acted on their political priorities, not those of the Iraqis," a government statement said.

The attacks "force Iraq to review its relations and its security, political and legal framework to protect its sovereignty", it added.